# X-ray Diffraction of Cubic structures ```X-ray Diffraction (XRD)
Learning Objectives


To learn the operation of a x-ray diffraction system
To obtain a 2Theta - Intensity (I) scan and index the peaks to identify the crystal
structure and its lattice parameters
Theory
Creation of X-ray’s:
When a high-energy electron beam (10-40KV) hits a water-cooled metallic target
such as Cu or Mo, the targets emits x-rays corresponding to energies of electronic
transitions within the atoms. For example, Cu k (α) = 1.54 A and Mo k (α) = 1.54 A.
The x-rays can be made highly monochromatic (very small bandwidth) by using
monochromators.
What is x-ray diffraction?
When the x-rays (E-M wave of low wavelength) impinge on a material, the x-rays
interact with the material’s atomic core and the electronic cloud around the atoms. Since
the x-ray is an electro-magnetic radiation, it gets scattered by the atoms. The strength of
scattering is dependent on the atom itself, i.e., atomic number, Z, and is called the atomic
scattering factor.
Any solid materials can be classified as single crystal, polycrystalline or
amorphous. Depending on the crystalline of the material, the scattered x-rays will
interfere, either constructively (crystals) or destructively (amorphous materials). The
phenomenon of interference is called diffraction. Diffraction will result only when the
wavelength is comparable to or smaller than the distance between the scatterers, i.e.,
atoms. The diffraction pattern is usually a 3-D pattern and is related to the crystal lattice
in a reciprocal way. Actually, it can be shown that the reciprocal lattice (diffraction
pattern) is related to a real crystal lattice through Fourier transform, exactly the same way
a time domain signal is related to the frequency domain representation.
Procedure
X-ray diffraction will be conducted on samples (in the Geosciences department) and the
data will be given to you in the form of 2Theta vs Intensity (I) scans or data file. You will
visit the physics department and learn how it is done.
General procedure for indexing and obtaining crystal structures
 Obtain the 2Theta values corresponding to each peak and tabulate it.
 Corresponding to each 2Theta value, calculate and tabulate the interplanar
distance, dhkl, using Bragg’s law of diffraction:
d = λ/ 2Sinθ
where: λ is the wavelength of the x-ray.



Normalize all the dhkl values to the first one and hence the first one has a unity
value.
Now compare the normalized (explained in the previous step) value from the
experimental data to the normalized data from the structure factor calculation for
the various structures (Will be provided by the TA). One set of the structure
factor calculation will match with the experimental value. Pick that crystal as
structure of the material.
Calculate the lattice constant from Bragg’s law for the various hkl’s and average
them.
Questions
1.) What are the crystal structures (BCC, FCC, HCP, etc…) of Al, Cu, Mg, Zn, KCl,
NaCl, and RbCl?
2.) Calculate the lattice constants of these materials?
3.) How would you proceed if the sample is a mixture of materials such as a metal (Al)
and a semiconductor (Si)?
```